Similarity and Transformations Similarity Warm Up Lesson Presentation

Similarity and Transformations Similarity Warm Up Lesson Presentation Lesson Quiz Holt. Geometry Mc. Dougal Geometry

Similarity and Transformations Warm Up Find the image point when the indicated transformation is applied to the given pre-image point. 1. (x, y) → (x + 3, y – 1); (2, 4) (5, 3) 2. (x, y) → (x, –y); (– 2, 1) (-2, -1) 3. (x, y) → (3 x, 3 y); (– 5, 0) (-15, -0) 4. (x, y) → 1 x, 1 y ; (3, -6) 3 3 (1, -2) Holt Mc. Dougal Geometry

Similarity and Transformations Objectives Draw and describe similarity transformations in the coordinate plane. Use properties of similarity transformations to determine whether polygons are similar and to prove circles similar. Holt Mc. Dougal Geometry

Similarity and Transformations Vocabulary similarity transformation Holt Mc. Dougal Geometry

Similarity and Transformations A transformation that produces similar figures is a similarity transformation. A similarity transformation is a dilation or a composite of one or more dilations and one or more congruence transformations. Two figures are similar if and only if there is a similarity transformation that maps one figure to the other figure. Holt Mc. Dougal Geometry

Similarity and Transformations Remember! Translations, reflections, and rotations are congruence transformations. Holt Mc. Dougal Geometry

Similarity and Transformations Example 1: Drawing and Describing Dilations A. Apply the dilation D to the polygon with the given vertices. Describe the dilation. D: (x, y) → (3 x, 3 y) A(1, 1), B(3, 1), C(3, 2) dilation with center (0, 0) and scale factor 3 Holt Mc. Dougal Geometry

Similarity and Transformations Example 1: Continued B. Apply the dilation D to the polygon with the given vertices. Describe the dilation. D: (x, y) → 3 x, 3 y 4 4 P(– 8, 4), Q(– 4, 8), R(4, 4) 3 dilation with center (0, 0) and scale factor 4 Holt Mc. Dougal Geometry

Similarity and Transformations Check It Out! Example 1 1 x, 1 y to the Apply the dilation D : (x, y)→ 4 4 polygon with vertices D(-8, 0), E(-8, -4), and F(4, -8). Name the coordinates of the image points. Describe the dilation. D'(-2, 0), E'(-2, -1), F'(-1, -2); dilation with center 1 (0, 0) and scale factor 4 Holt Mc. Dougal Geometry

Similarity and Transformations Example 2 : Determining Whether Polygons are Similar Determine whether the polygons with the given vertices are similar. A. A(– 6, -6), B(-6, 3), C(3, 3), D(3, -6) and H(-2, -2), J(-2, 1), K(1, 1), L(1, -2) Yes; ABCD maps to HJKL by a dilation: 1 x, 1 y (x, y) → 3 3 Holt Mc. Dougal Geometry

Similarity and Transformations Example 2: Continued B. P(2, 0), Q(2, 4), R(4, 4), S(4, 0) and W(5, 0), X(5, 10), Y(8, 10), Z(8, 0). No; (x, y) → (2. 5 x, 2. 5 y) maps P to W, but not S to Z. Holt Mc. Dougal Geometry

Similarity and Transformations Example 2: Continued C. A(1, 2), B(2, 2), C(1, 4) and D(4, -6), E(6, -6), F(4, -2) Yes; ABC maps to A’B’C’ by a translation: (x, y) → (x + 1, y - 5). Then A’B’C’ maps to DEF by a dilation: (x, y) → (2 x, 2 y). Holt Mc. Dougal Geometry

Similarity and Transformations Example 2: Continued D. F(3, 3), G(3, 6), H(9, 3), J(9, – 3) and S(– 1, 1), T(– 1, 2), U(– 3, 1), V(– 3, – 1). Yes; FGHJ maps to F’G’H’J’ by a reflection : (x, y) → (-x, y). Then F’G’H’J’ maps to STUV by a dilation: (x, y) 1 x , 1 y 3 3 Holt Mc. Dougal Geometry

Similarity and Transformations Check It Out! Example 2 Determine whether the polygons with the given vertices are similar : A(2, -1), B(3, -1), C(3, -4) and P(3, 6), Q(3, 9), R(12, 9). The triangles are similar because ABC can be mapped to A'B'C' by a rotation: (x, y) → (-y, x), and then A'B'C' can be mapped to PQR by a dilation: (x, y) → (3 x, 3 y). Holt Mc. Dougal Geometry

Similarity and Transformations Example 3: Proving Circles Similar A. Circle A with center (0, 0) and radius 1 is similar to circle B with center (0, 6) and radius 3. Holt Mc. Dougal Geometry

Similarity and Transformations Example 3: Continued Circle A can be mapped to circle A’ by a translation: (x, y) → (x, y + 6). Circle A’ and circle B both have center (0, 6). Then circle A’ can be mapped to circle B by a dilation with center (0, 6) and scale factor 3. So circle A and circle B are similar. B. Circle C with center (0, – 3) and radius 2 is similar to circle D with center (5, 1) and radius 5. Holt Mc. Dougal Geometry

Similarity and Transformations Example 3: Continued Circle C can be mapped to circle C’ by a translation: (x, y) → (x + 5, y + 4). Circle C’ and circle D both have center (5, 1). Then circle C’ can be mapped to circle D by a dilation with center (5, 1) and scale factor 2. 5. So circle C and circle D are similar. Holt Mc. Dougal Geometry

Similarity and Transformations Check It Out! Example 3 Prove that circle A with center (2, 1) and radius 4 is similar to circle B with center (-1, -1) and radius 2. Circle A can be mapped to circle A' by a translation: (x, y) → (x - 3, y - 2). Then circle A' can be mapped to circle B by a dilation with center (-1, -1) and scale factor 1 So, circles A and B are 2 similar. Holt Mc. Dougal Geometry

Similarity and Transformations Example 4: Application Eric wants to make a drawing to show he will arrange three rugs in his room. The large rug is twice the size of each small rug. Eric will first draw the small rug in the lower left corner and then the large rug in the middle. How can he draw those rugs? Holt Mc. Dougal Geometry

Similarity and Transformations Example 4: Continue Place the lower left rug in the first quadrant of a coordinate plane in a convenient position. Apply the dilation with center (0, 0) and scale factor 2: (x, y) → (2 x, 2 y). Holt Mc. Dougal Geometry

Similarity and Transformations Check It Out! Example 4 What if…? How could Tia draw the middle flag to make it 4 times the size of each of the other flags? Apply the dilation with center (0, 0) and scale factor 4: (x, y) → (4 x, 4 y). Holt Mc. Dougal Geometry

Similarity and Transformations Lesson Quiz : Part-I 1. Apply the dilation D: (x, y) to the polygon with vertices A(2, 4), B(2, 6), and C(6, 4). Name the coordinates of the image points. Describe the dilation. A’(3, 6), B’(3, 9), C’(9, 6); dilation with center 3 (0, 0) and scale factor 2 Holt Mc. Dougal Geometry

Similarity and Transformations Lesson Quiz : Part-II Determine whether the polygons with the given vertices are similar. 2. A(-4, 4), B(6, 4), C(6, -4), D(-4, -4) and P(-2, 2), Q(4, 2), R(4, -2), S(-2, -2) No; (x, y) → (0. 5 x, 0. 5 y) maps A to P, but not B to Q. 3. A(2, 2), B(2, 4), C(6, 4) and D(3, -3), E(3, -6), F(9, -6) Yes; △ ABC maps to △ A’B’C’ by a reflection: (x, y) → (x, -y). Then △ A’B’C’ maps to △DEF by a dilation: (x, y) → (1. 5 x, 1. 5 y). Holt Mc. Dougal Geometry

Similarity and Transformations Lesson Quiz : Part-III 4. Prove that circle A with center (0, 4) and radius 4 is similar to circle B with center (-2, -7) and radius 6. Circle A can be mapped to circle A’ by a translation: (x, y) → (x - 2, y - 11). Circle A’ and circle B both have center (-2, -7). Then circle A’ can be mapped to circle B by a dilation with center (-2, -7) and scale factor 3 So circle A and circle B are similar. 2 Holt Mc. Dougal Geometry